A209200 G.f.: (1-4*x)^(-1/2) * (1-8*x)^(-1/4).

1, 4, 20, 112, 680, 4384, 29536, 205440, 1462368, 10587520, 77633920, 574845440, 4289409280, 32206976000, 243074083840, 1842511532032, 14018197145088, 106996519311360, 818973463721984, 6284217844736000, 48327723087278080, 372397083591557120

Offset: 0

Author

Paul D. Hanna, Mar 06 2012

Keywords

• nonn

Comments

Equals the convolution of sequences A000984 and A004981.

The sequences A000984 and A004981 are related by the identity:

Sum_{n>=0} A000984(n)^3 *x^n = ( Sum_{n>=0} A004981(n)^2 *x^n )^2.

Examples

			G.f.: A(x) = 1 + 4*x + 60*x^2 + 1200*x^3 + 27300*x^4 + 668304*x^5 +...
This sequence equals the convolution of the sequences:
A000984 = [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, ...], and
A004981 = [1, 2, 10, 60, 390, 2652, 18564, 132600, 961350, ...].
Related sequences:
A^2: [1, 8, 56, 384, 2656, 18688, 133888, 974848, 7194112, ...],
A^4: [1, 16, 176, 1664, 14592, 122880, 1011712, 8224768, ...].
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000984, A004981.

Programs

Mathematica

CoefficientList[Series[(1-4*x)^(-1/2)*(1-8*x)^(-1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

PARI

{a(n)=polcoeff((1-4*x +x*O(x^n))^(-1/2)*(1-8*x +x*O(x^n))^(-1/4),n)}

PARI

{A000984(n)=polcoeff((1-4*x +x*O(x^n))^(-1/2),n)}
{A004981(n)=polcoeff((1-8*x +x*O(x^n))^(-1/4),n)}
{a(n)=sum(k=0,n,A000984(n-k)*A004981(k))}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} A000984(n-k)*A004981(k).

Recurrence: n*a(n) = 4*(3*n-2)*a(n-1) - 8*(4*n-5)*a(n-2). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ Gamma(3/4)*8^n/(Pi*n^(3/4)). - Vaclav Kotesovec, Oct 20 2012